Hume-Rothery rules  were used for decades in traditional alloy design, as they postulate the conditions for substitutional solid solubility in binary systems, that is, the same crystal structure and valence state, similar electronegativity, and a difference in atom sizes between solute and solvent atoms that should be less than 15%.
Mizutani and his collaborators carried out an in-depth revision of the Hume-Rothery rules [6-8].
Mizutani, Hume-Rothery Rules for Structurally Complex Alloy Phases, CRC Press, Boca Raton, Fla, USA, 2010.
For some 90 years, explains Mizutani (Nagoy Industrial Science Research Institute, Japan), the Hume-Rothery rules
on factors that enhance or hinder the stability of phases in materials has focused attention on three aspects of alloying: the electrochemical effects, the size factor effects, and the change of electron concentration.
And this does not fulfill the criteria for formation of solid solutions according to Hume-Rothery rules .
 Tsai, A., 2004, "A test of Hume-Rothery rules for stable quasicrystals," J.
In case the [Al.sub.2][O.sub.3] ceramic was used as support, the [Al.sub.2][O.sub.3] ceramic did not satisfy the Hume-Rothery rules for a homogeneous solid solution with Pd metal because of the poor chemical affinity between [Al.sub.2][O.sub.3] and Pd.
These main components exactly satisfied the requirements to form a homogeneous solid solution, as suggested by the Hume-Rothery rules [12, 22, 23]; Pd and Ni metals with high chemical affinity are completely soluble in the solid state over the entire compositional range at 923 K, as illustrated by the Pd-Ni phase diagram given in Figure 2(d) .